Recently I saw some physical problems that can be modeled by equations with fractional derivatives, and I had some doubts: is it possible to write an action that results in an equation with fractional derivatives? For example, consider a hypothetical physical system with the principle of least action. Is there a "wave equation" with the time-derivative $3/2$? Does such a question make sense?
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$\begingroup$ Out of curiosity, what sort of physical problems work this way? $\endgroup$– spencer nelsonJan 31, 2011 at 20:56
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$\begingroup$ @Spencer I saw the first time in anomalous diffusion. You can see here, for example: pfi.uem.br/mfi/disserta_teses/teses_pdf/… (in portuguese). This is a investigation of a system governed by a non-Markovian Fokker-Planck equation that are related with the comb model. $\endgroup$– gsAllanFeb 1, 2011 at 1:26
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$\begingroup$ Forces proportional to velocity, such as friction for example, can be described by putting fractional derivatives in the Lagrangian. $\endgroup$– SMeznaricJul 5, 2013 at 23:11
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1$\begingroup$ Here is the tome our group did a lot of work out of: Metzler, R. (2000). The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339(1), 1–77. doi:10.1016/S0370-1573(00)00070-3. It is a great introduction to the subject. There should be an arxiv preprint available but I don't have the number handy. $\endgroup$– MichaelJul 6, 2013 at 0:14
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2 Answers
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Fractional derivatives are nonlocal, but actions are usually assumed to be local.
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$\begingroup$ the locality or non-locality of a Lagrangian density depends on the highest order of derivatives in its various terms. Terms with upto second order derivative (eg. kinetic terms) are local. Terms with derivatives of order 3 and higher are non-local, with the degree of non-locality being measured by the order. So a non-local term in an action does not imply that the variational problem is ill-defined. Also why are fractional derivatives non-local? Even better, what is a fractional derivative? Thanks. $\endgroup$– user346Jan 27, 2011 at 17:01
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4$\begingroup$ Dear @space_cadet, no, what you write is not correct. QGR is right that whenever fractional derivatives appear, the action is nonlocal. It's enough for some terms to appear, whether they're leading or not, and the action has to be nonlocal. At the end of your answer, you admit that you don't really know why they're nonlocal. Well, it's because the you may write $\partial_t^k$ as $E^k$ in the Fourier transformed energy representation, and the transformation of the operator $E^k$ back to the $t$ basis is nonlocal. $\endgroup$ Jan 27, 2011 at 17:19
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$\begingroup$ @space_cadet sloppy physicist thinking here, but I think it can be made precise: imagine taking a taylor series of the fractional derivative --- it would need to go to infinite order, and thus it would be highly non-local. $\endgroup$– gennethJan 28, 2011 at 10:43
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$\begingroup$ @space_cadet, a fraction derivative is a simple example of "pseudo-differential operator". It's an operator that can be easily defined for example with fourier transform (like Lubos Motl sayed). You can define operator that in Fourier Transform is equivalent to $\sin{k}$...Fraction derivative is simpler, in Fourier Transform is a multiplication for your transform variable $k$ with fractional exponent ;-) $\endgroup$– Boy SJan 28, 2011 at 12:36
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1$\begingroup$ @Boy, @genneth - thanks. @Lubos, all I was saying is that, AFAIK, having non-local terms is fairly common, eg. higher derivative theories of gravity. So being non-local should not be a obstacle for an action with fractional derivatives, as @QGR seems to suggest. $\endgroup$– user346Jan 28, 2011 at 12:40
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When I've seen fractional derivatives I've assumed that one place where they would naturally arise is in physical situations where there's a fractional dependency on time.
For example, random walks typically result in movement proportional to $\sqrt{t}$. Googling for "fractional+derivative+random+walk" on arxiv.org finds some papers that explore this:
http://www.google.com/search?q=fractional+derivative+random+walk+site%3Aarxiv.org
So I'm wondering if there's a way of relating some of the diffusion versions of QM with fractional derivatives.
answered Feb 1, 2011 at 0:40